Papers

It is a classical result dating back to Dehn (1903) that squares tiling a rectangle must have rational side lengths. However, bounding the arithmetic complexity of these tilings, specifically the growth of their denominators, is an active area of research. Recent contributions have tackled these arithmetic constraints through the complexity of planar multigraphs and Diophantine approximations. In this paper, we offer an algebraic perspective of this problem by encoding the tiling's geometry into a block matrix. This approach allows us to reduce geometric constraints to a finite set of additive relations. By exploiting the total unimodularity of interval matrices (Ghouila-Houri, 1962), we provide a self-contained proof that the least common multiple of the denominators is bounded by $2^n$ in two dimensions, where $n$ is the order of the tiling. Furthermore, because we can naturally generalizes this matrix to represent $d$-dimensional hypercube tilings, yielding a bound of $d^n$. We demonstrate the utility of this framework by providing a brief proof of Richard Kenyon's 1996 theorem regarding the minimum number of squares required to tile an integer rectangle.

In the past two years, several points of view have been proposed to address the question of the generalization of the theory of free probability to random tensors with different invariances, and it is unclear at this point whether they lead to the same notions of tensorial free cumulants and freeness. One way to approach this problem, developed by Collins, Gurau and the second named author for local unitary invariant random tensors, relies on finite size quantities involving averages over the invariance group, and whose asymptotics naturally possess the properties expected for tensorial generalizations of free cumulants of arbitrary orders. At this point, this approach has only been carried out for certain distributions, and for a subset of the moments that define such theories, and a more systematic and exhaustive study is lacking. This is the program initiated in this paper: we link this approach to the one proposed by Nechita and Park; extend a number of their results as well as those of the aforementioned paper to arbitrary orders of fluctuations, thereby generalizing higher order free cumulants; push further the study of distributions with larger invariance groups; detail the link with the asymptotics of the free-energies of the tensor HCIZ and BGW integrals; and provide formulae for tensorial free cumulants of products of tensors. Another important question is that of the definition of concrete distributions whose tensorial free-cumulants take non-trivial values. We compute the tensorial free cumulants for Gaussian random tensors with non-trivial covariances, and show that they provide such examples.

A valuation-theoretic framework is developed for studying tropical degenerations of multilinear network games. Equilibrium conditions are modeled by an ideal over the Puiseux field, and valuation classes and cluster multiplicities are used to describe the organization of Puiseux equilibrium branches under degeneration. For valuation vectors lying in the relative interiors of generator-wise maximal tropical cells, multilinearity is shown to force a binomial reduction of the generator-wise initial system. The resulting binomial systems are governed by exponent-difference graphs, strongly connected component decompositions, and lattice indices computed via Smith normal form. In particular, unimodular diagonal blocks yield initial-coefficient rigidity, whereas non-unimodular blocks give rise to torsion-type leading-coefficient multiplicities. The generic binomial theory is complemented by a collision-normalized cross-prism family. In this family, 2^L Puiseux equilibrium branches share the same valuation vector and the same leading-coefficient vector. The corresponding limiting initial fiber is supported at a single torus point, but this point is nonreduced with scheme-theoretic length 2^L. Thus valuation coalescence is realized as an intrinsic scheme-theoretic collision rather than merely as a loss of higher-order terms. These local degeneration invariants are then related to the algebraic-degree theory of network games, showing how global equilibrium counts can be refined by valuation classes, binomial initial systems, Smith lattice data, and nonreduced collision fibers.

A $k$-plane of a $d$-dimensional array is a subarray formed by fixing $d-k$ coordinates and allowing the remaining $k$ coordinates to vary freely. A Latin hypercube of dimension $d$ and order $n$ is an $n\times n\times\cdots\times n$ array of dimension $d$ containing symbols from an $n$-set, such that each $1$-plane contains each of the possible entries exactly once. A transversal in a Latin hypercube of order $n$ is a set of $n$ entries of the hypercube, no pair of which agree in any coordinate or contain the same symbol. The aim of this paper is to construct Latin hypercubes that have transversals but which have many entries that are not in any transversal, or for which the number of disjoint transversals is limited. We show the following results in the case when the dimension $d$ is even. For all even $n\ge 10$ there exists a Latin hypercube of order $n$ that contains a transversal but for which all transversals hit one $(d-2)$-plane. For $n\in\{6,8\}$ there exists a Latin hypercube of order $n$ that contains a transversal but for which all transversals hit one of two $(d-2)$-planes. For even $d>2$ there is a Latin hypercube of order $n=4$ that contains a transversal but has $2^d$ entries that are not in any transversal. Our constructions use a quasigroup $(Q,\ast)$ to increase the dimension of a Latin hypercube using the rule $H_d(x_1,\dots,x_d)=H_{d-1}(x_1,\dots,x_{d-1})\ast x_d$. We give several characterisations which allow us to diagnose which entries of $H_d$ are in transversals in terms of properties of $H_{d-1}$ and $Q$.

Given a graph $F$, the $r$-expansion $F^{(r)+}$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Recently, Alon and Frankl (JCTB, 2024) and Gerbner (JGT, 2023) studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. Gerbner, Tompkins and Zhou (EJC, 2025) considered the analogous Turán problems on hypergraphs with bounded matching number. In this paper, we study hypergraph extensions of the Alon–Frankl Theorem. More precisely, we determine the maximum number of hyperedges in an $n$-vertex $r$-uniform hypergraph containing neither a matching $M^r_{s+1}$ nor the expansion $K_{\ell+1}^{(r)+}$ of the clique $K_{\ell+1}$ for all small $s<\frac{\ell^2-1}{2}$ and all sufficiently large $s$, respectively. This result partly confirms a conjecture proposed by Gerbner, Tompkins and Zhou (EJC, 2025). As a key tool, we determine the rainbow hyper-Turán number for expansions of cliques, which is defined as the maximum sum of size of a sequence of hypergraphs $\mathcal{H}_1,\dots,\mathcal{H}_k$ that contains no rainbow copies of expansions of cliques with given size. It extends the result of Keevash, Saks, Sudakov and Verstraëte (AAM, 2004), which determined the rainbow Turán number of cliques in the graph case. These results shows a correlation between the hyper-Turán problem and the rainbow hyper-Turán number.

Wang, Cui, and Cioabă introduced the Laplacian matching root integral variation of a graph and proved that it cannot occur in one place. They also showed that the two-place variation is impossible for connected graphs satisfying $g(G)/c(G)>7/6$, where $g(G)$ is the girth and $c(G)$ is the dimension of the cycle space, and conjectured that no connected graph admits such a two-place variation. In this paper, we confirm this conjecture. The proof combines a structural relation obtained in their paper with two new power-sum identities for Laplacian matching roots.

We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.

Determining an upper bound on $s$ for vertex-primitive $s$-arc-transitive digraphs has been an open problem of considerable interest since a question asked by Praeger in 1990. Although much progress has been made and an upper bound is conjectured to be $2$, a complete classification for $s=2$ remains out of reach. In this paper, we prove that the tight upper bound on $s$ for finite vertex-primitive $s$-arc-transitive Cayley digraphs is exactly $2$. Furthermore, we completely characterize the structure of these digraphs when $s=2$.

Fix $k \in \mathbb{N}$ and let $G$ be a connected graph with treewidth at most $k$. We say that $xy \notin E(G)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T, \cB)$ of $G$ with width at most $k$, both $x$ and $y$ are contained in a bag of $(T, \cB)$. Moreover, if $G$ does not contain any $k$-ghost-edges, then $G$ is {\em $k$-ghost-free}. Hickingbotham proposed a conjecture that every connected $k$-ghost-free graph $G$ has a tree decomposition $(T, \cB)$ with width at most $k$ such that $T$ is a subgraph of $G$. In this paper, we prove that Hickingbotham's conjecture is false for all $k\geq3$.

We introduce the Banach-Butterfly Invariant (BBT), an influence-adaptive Banach geometry on the Walsh-Hadamard butterfly factorization. For a Boolean function $f:\{-1,+1\}^n\to\{-1,+1\}$ with coordinate influences $\mathrm{Inf}_\ell(f)$, BBT assigns exponent $p_\ell = 1+\mathrm{Inf}_\ell(f)$ to butterfly layer $\ell$, yielding the contraction invariant $\mu(f)=\prod_\ell 2^{-\mathrm{Inf}_\ell/(1+\mathrm{Inf}_\ell)}$. We prove a Jensen lower bound $\log_2\mu(f) \ge -I(f)/(1+I(f)/n)$ and that $\mu$ is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes $\mu\sim 2^{-n/2}$ (parity), $2^{-\Theta(\sqrt{n})}$ (majority), $2^{-1/2}$ (dictators). $\log_2\mu$ is rational but not polynomial in the Fourier coefficients while $\mu$ is algebraic, and $\mu$ separates functions with identical total influence (122 pairs at $n=3$). Using the certified $n \le 4$ ternary Walsh-threshold universe from a companion synthesis manuscript as a finite testbed, we compute exact MILP minimum-support certificates for all 65,536 Boolean functions at $n=4$ (mean 6.42, max 9, all-odd by a parity argument) and on 10,000 of the 616,126 NPN-canonical representatives we enumerate at $n=5$ (matching OEIS A000370). Conditional Spearman $\rho(\mu,|\mathrm{supp}|)$ at fixed total influence is $+0.571$ in the largest stratum at $n=4$ but reverses to $-0.38$ at $n=5$ under both function-uniform and NPN-canonical sampling: $\mu$ is a valid Schur-convex concentration invariant, not a universal monotone predictor of minimum support across $n$. A companion application paper validates a real-valued WHT activation-energy proxy inspired by this theory on five pretrained LLMs at W2A16, cutting wikitext-2 perplexity by 15-58% versus vanilla auto-round; the transfer from Boolean theory to the real-valued proxy is qualitative, not formal.

We resolve a conjecture posed by Rauzy in 1983 concerning the $d$-ary generalizations of Sturmian words. A classical theorem by Coven and Hedlund from 1973 states that Sturmian words, which are classically defined by their subword complexity (a combinatorial refinement of topological entropy), or as the natural codings of irrational rotations on the circle, are also characterized by their abelian complexity (a combinatorial refinement of the notion of discrepancy). More precisely, Sturmian words are exactly the binary infinite words with rationally independent letter frequencies and minimal abelian complexity, which is in this case constant and equal to $2$. We prove that there exist no infinite ternary words with rationally independent letter frequencies and constant abelian complexity equal to $3$, thereby establishing Rauzy's conjecture.

This article settles Problem 7.2 posed by [Banerjee, Special Matrices (2022)] for the induced subgraph $G_2$ of the comaximal graph $\Gamma(\mathbb Z_n)$ when $n$ is squarefree. Let $n=p_1p_2\cdots p_m$ with distinct primes $p_1<\cdots<p_m$, and let $G_2$ be the graph on the nonzero nonunit residue classes modulo $n$. We use Chinese remainder representation of $\mathbb Z_n$, and encodes each vertex by the set of vanishing coordinates. This converts $G_2$ into a weighted blow-up of a disjointness graph on nonempty proper subsets of $\{1,\dots,m\}$. Within this model, we derive exact class sizes, explicit degree formulas, the minimum-degree layer, and a short-path criterion. The main theorem proves the connectivity of $G_{2}$ as $\kappa(G_2)=\prod_{i=1}^{m-1}(p_i-1)=\tfrac{\phi(n)}{p_m-1}$. Consequently, earlier upper bound is sharp, $G_2$ is maximally connected, and its edge connectivity agrees with its minimum degree. We also obtain distance formulas, diameter and radius information, and a linear-time algorithm once the prime factorization is known.

The chain length of a set family $\mathcal{S} \subseteq 2^{[m]}$ is the largest ascending sequence of sets in containment order in the union-closure of $\mathcal S$. In this work, we provide a significantly simpler and more optimal characterization of the sparsifiability of set systems in terms of their chain length, improving on the work of Brakensiek and Guruswami [STOC 2025]. Our proof relies on a generalization of Karger's [SODA 1993] famous contraction algorithm and its recent linear algebraic extensions [Khanna-Putterman-Sudan SODA 2024], and our resulting bounds show that, just as VC dimension characterizes the additive sparsifiability of a set system, chain length governs the multiplicative sparsifiability. As a corollary, we obtain improved bounds for weighted CSP sparsification.

We study a variant of the classical Wythoff's game. The classical form is played with two piles of stones, from which two players take turns to remove stones from one or both piles. When removing stones from both piles, an equal number must be removed from each. The player who removes the last stone or stones is the winner. Equivalently, we consider a single chess queen placed somewhere on a large grid of squares. Each player can move the queen toward the upper-left corner of the grid, either vertically, horizontally, or diagonally in any number of steps. The winner is the player who moves the queen to the terminal position in the upper-left corner, the position (0,0) in our coordinate system. Let k be a positive integer, and we consider the variant of Wythoff's game with the terminal set {(x,y):x,y are non-negative integers and x+y <=k}. The set of P-positions of this variant is described by the Fibonacci sequence without using recursion.

We approach the cycle double cover conjecture by looking for a circular 2-cell embedding of cubic graphs on an arbitrary surface. It is easy to see that if such an embedding exists, we can get to it from an arbitrary starting 2-cell embedding by repeating “twists of an edge”. We study this twisting operation in detail and deduce bounds on the number of singular edges (edges where a face meets itself).

Let $p$ be an odd rational prime and consider the cyclotomic number field $K = \mathbb{Q}(\zeta_{p})$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen–Franks group is closely related to the minus part of the class group of $K$. In particular, both groups have the same cardinality up to an explicit power of $p$. Furthermore, they are both $\mathrm{Gal}(K/\mathbb{Q})$-modules, and we prove the equality of the cardinalities of their isotypic components after tensoring them with the valuation ring of an appropriate $\ell$-adic field for $\ell \nmid p-1$.

In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit description of the fan of a toric Schubert variety, leading to a combinatorial model for its cones. As an application, we obtain necessary and sufficient conditions for smoothness of toric Schubert varieties in terms of the Cartan integers associated to a reduced expression. Furthermore, we prove that for a Coxeter-type element $w \in W^P$, the interval $[e,w]_{W^P}$ is a supersolvable join-distributive lattice. Finally, we apply these results to the study of spherical and horospherical Schubert varieties, providing a combinatorial method for checking the smoothness via the associated toric Schubert varieties.

A problem originating with Erdős and Silverman in the 1970s asks for the minimum integer $r(k)$ such that any set of $n \ge r(k)$ points in the plane has some $k$-subset with no right angles. The case $k=4$ has an interesting gap between the known bounds, namely $8 \le r(4) \le 10$. Here, we consider a relaxation that quantifies the deviation from right angles. Specifically, we study $\Gamma_k(n)$, the supremum of angles $\gamma$ such that every $n$-set of points in $\mathbb{R}^2$ has a $k$-subset with all angles outside of the interval $90^\circ \pm \gamma$. We show that $4^\circ \le \Gamma_4(10) \le 9.292^\circ$. For large $n$, the quantity $\Gamma_3(n)$ is closely related to a classical minimax angle problem pioneered by Blumenthal, Erdős and Szekeres. We give bounds on $\Gamma_k(n)$ for a general $k$ and large $n$.

Let $\mathcal{D}$ be a family of digraphs. A digraph $D$ is \emph{$\mathcal{D}$-saturated} if it contains no member of $\mathcal{D}$ as a subdigraph, but for any arc $e$ in the complement of $D$, the digraph $D + e$ contains some member of $\mathcal{D}$ as a subdigraph. The saturation number $\mathrm{sat}(n,\mathcal{D})$ and the extremal number $\mathrm{ex}(n,\mathcal{D})$ are the minimum number and the maximum number of arcs among all $n$-vertex $\mathcal{D}$-saturated digraphs. For a positive integer $k$, let $\mathcal{D}_k$ denote the family of \emph{$k$-strongly connected digraphs}. In this paper, firstly, we prove that $$\mathrm{sat}(n,\mathcal{D}_k)=(k-1)(2n-k)+\binom{n-k+1}{2}.$$ Then for $n\geq 3(k-1)$, we prove that $$\mathrm{ex}(n,\mathcal{D}_k)\leq \binom{n-k+1}{2}+\frac{17}{6}(k-1)(n-k+1).$$ In addition, we conjecture that for sufficiently large $n$, $$\mathrm{ex}(n,\mathcal{D}_k)=\binom{n}{2}+\frac{3}{2}(k-\frac{4}{3})(n-k+1).$$

The Domination game is an impartial game on graphs, introduced in 2010, and proved PSPACE-complete in the normal variant in 2026. In this game, Alice and Bob alternately select playable vertices, where a vertex is playable if it dominates at least one vertex not dominated by the vertices selected before in the game. The game ends when the selected vertices form a dominating set. In the normal variant, the player unable to move loses. In contrast to the impartial game, the partizan game has the vertices already colored with $A$, $B$, or $C$, in such a way that Alice (resp. Bob) can only select vertices colored with $A$ (resp. $B$) or $C$. The partizan game was proved PSPACE-hard in 2026. In this paper, we determine the winner of the Normal Domination Partizan game in graphs whose connected components are complete bipartite graphs or complete split graphs, including star forests, for any initial coloring of its vertices.

We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$\alpha$ if and only if it is "$(\mathrm{tw},\omega)$-bounded" (that is, for all $t\in \mathbb N$, the class of all $K_t$-free graphs in $\mathcal{G}$ has bounded treewidth). Equivalently, $\mathcal{G}$ has bounded tree-$\alpha$ if and only if it excludes a complete bipartite graph, a forest whose components each have at most three leaves, and the line graph of such a forest. This resolves two conjectures of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht: the above, and a weaker one that for all $a,b\in \mathbb N$, every hereditary class that excludes $K_{a,a}$ and the $b$-vertex path has bounded tree-$\alpha$. The latter was already open even for $(a,b)\in \{(2,7),(3,5)\}$, and only recently proved for $(a,b)=(2,6)$.

LDPC codes play a vital role in coding theory and practical error correction. A central problem in this direction is to understand their rate–distance tradeoff. In this paper, we develop a local-growth argument for estimating ball sizes in the coset graphs of LDPC codes. Rather than estimating coset balls directly, we use a local-growth analysis to bound the coset-weight generating function of linear spaces spanned by low-weight vectors. This approach sharpens the previous ball-size estimates of Iceland and Samorodnitsky. Combined with a general method of Friedman and Tillich that relates balls in coset graphs to sizes of error-correcting codes, it further improves the upper bounds on the rate of LDPC codes for a significant range of relative distances.

We give a complete characterization of generic irreducibility for dispersion polynomials and Bloch varieties of periodic graph operators. More precisely, we prove that for a generic choice of edge weights and potentials, the dispersion polynomial/Bloch variety of a nontrivial periodic graph is irreducible if and only if the quotient graph is connected. Our proof uses a strong dichotomy for parameterized Laurent polynomials: reducibility either occurs for every parameter or fails on a nonempty Zariski-open set. After establishing this dichotomy, we reduce the problem to minimally connected periodic graphs.